TY - JOUR

T1 - Elliptic-regularization of nonpotential perturbations of doubly-nonlinear flows of nonconvex energies

T2 - A variational approach

AU - Akagi, Goro

AU - Melchionna, Stefano

N1 - Funding Information:
G.A. is supported by JSPS KAKENHI Grant Number 16H03946 and by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation. S.M. is supported by the Austrian Science Fund (FWF) project P27052-N25. The Authors would like to acknowledge the kind hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics, where part of this research was developed under the frame of the Thematic Program Nonlinear Flows.
Publisher Copyright:
© 2018 Heldermann Verlag. All rights reserved.

PY - 2018

Y1 - 2018

N2 - This paper presents a variational approach to doubly-nonlinear (gradient) flows (P) of nonconvex energies along with nonpotential perturbations (i.e., perturbation terms without any potential structures). An elliptic-in-time regularization of the original equation (P)ϵ is introduced, and then, a variational approach and a fixed-point argument are employed to prove existence of strong solutions to (P)ϵ. More precisely, we introduce a family of functionals (defined over entire trajectories) parametrized by a small parameter ϵ, whose Euler-Lagrange equation corresponds to the elliptic-in-time regularization of an unperturbed (i.e. without nonpotential perturbations) doubly-nonlinear flow. Secondly, due to the presence of nonpotential perturbation, a fixed-point argument is performed to construct strong solutions uϵ to the elliptic-in-time regularized equations (P)ϵ. Finally, a strong solution to the original equation (P) is obtained by passing to the limit of uϵ as ϵ → 0. Applications of the abstract theory developed in the present paper to concrete PDEs are also exhibited.

AB - This paper presents a variational approach to doubly-nonlinear (gradient) flows (P) of nonconvex energies along with nonpotential perturbations (i.e., perturbation terms without any potential structures). An elliptic-in-time regularization of the original equation (P)ϵ is introduced, and then, a variational approach and a fixed-point argument are employed to prove existence of strong solutions to (P)ϵ. More precisely, we introduce a family of functionals (defined over entire trajectories) parametrized by a small parameter ϵ, whose Euler-Lagrange equation corresponds to the elliptic-in-time regularization of an unperturbed (i.e. without nonpotential perturbations) doubly-nonlinear flow. Secondly, due to the presence of nonpotential perturbation, a fixed-point argument is performed to construct strong solutions uϵ to the elliptic-in-time regularized equations (P)ϵ. Finally, a strong solution to the original equation (P) is obtained by passing to the limit of uϵ as ϵ → 0. Applications of the abstract theory developed in the present paper to concrete PDEs are also exhibited.

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M3 - Article

AN - SCOPUS:85045915223

VL - 25

JO - Journal of Convex Analysis

JF - Journal of Convex Analysis

SN - 0944-6532

IS - 3

ER -